To factor the polynomial [tex]\( 3x^3 - 15x^2 + 18x \)[/tex], let's follow the step-by-step process.
1. Identify the common factor:
The polynomial [tex]\( 3x^3 - 15x^2 + 18x \)[/tex] has a common factor in each term. We can see that each term is divisible by [tex]\( 3x \)[/tex].
[tex]\[
3x^3 - 15x^2 + 18x = 3x(x^2 - 5x + 6)
\][/tex]
2. Factor the quadratic:
Next, we need to factor the quadratic expression [tex]\( x^2 - 5x + 6 \)[/tex]. To do this, we'll find two numbers that multiply to the constant term (6) and add up to the coefficient of the linear term (-5).
The pairs of numbers that multiply to 6 are:
- [tex]\( (1, 6) \)[/tex]
- [tex]\( (2, 3) \)[/tex]
Out of these pairs, [tex]\( -2 \)[/tex] and [tex]\( -3 \)[/tex] add up to [tex]\( -5 \)[/tex]:
[tex]\[
-2 \cdot -3 = 6 \quad \text{and} \quad -2 + (-3) = -5
\][/tex]
So, [tex]\( x^2 - 5x + 6 \)[/tex] factors to:
[tex]\[
(x - 2)(x - 3)
\][/tex]
3. Combine the factors:
Now, we substitute the factored quadratic expression back into the initial factor:
[tex]\[
3x(x^2 - 5x + 6) = 3x(x - 2)(x - 3)
\][/tex]
Therefore, the factorized form of the polynomial [tex]\( 3x^3 - 15x^2 + 18x \)[/tex] is:
[tex]\[
3x(x - 2)(x - 3)
\][/tex]
So, the correct answer is:
C. [tex]\( 3x(x-2)(x-3) \)[/tex]
